Abstract: In fuzzy reasoning, when the input and inference rules are perturbed, the selection of the internal connection operator is the main factor affecting the reasoning output. In this paper, the definition of fuzzy Lipschitz operator is given, the triangular- norm operators and implication operators satisfying Lipschitz condition are proved. The stable Lipschitz operators to influence the robustness of fuzzy reasoning are focused on. When the input is perturbed and the rule is perturbed, the internal connection operators which are 1-Lipschitz operators can effectively restrain the output perturbation, especially when the internal connective operators are 1-k∞-Lipschitz and quasi-copulas, the output of fuzzy reasoning is safer and more feasible, the robustness of fuzzy reasoning can get better control. In addition, the experimental results show that the rule perturbation has great effect on output. The experimental part is not only a good verification of the theory proposed in this paper, but also a specific application of the theory in image processing and face association.

Key words: Lipschitz, t-norm, implication, perturbation, fuzzy reasoning

摘要： 模糊推理中，输入和推理规则发生摄动时，内部连接算子的选择是影响推理输出的主要因素。给出了模糊Lipschitz聚合算子的定义，论证了满足Lipschitz条件的三角模算子和蕴涵算子，研究了一类稳定的Lipschitz聚合算子对模糊推理的鲁棒性影响，指出了当系统发生输入摄动和规则摄动时，内部连接算子为1-Lipschitz算子，能有效地抑制模糊推理的输出摄动，特别是当内部连接算子既是1-k∞-Lipschitz又是quasi-copulas时，模糊推理输出更稳定安全可行，模糊推理的鲁棒性得到更好调控；另外，从实验结果看，规则摄动对推理输出影响较大。实验部分既是对文中所提理论的很好验证，同时也是该理论在图像处理和人脸联想方面的具体应用。

关键词: Lipschitz, 三角模, 蕴涵, 摄动, 模糊推理